An ellipse has the following properties:
It is tangent to the $x$-axis at $(4,0).$
It is tangent to the $y$-axis at $(0,4).$
Its axes are parallel to the coordinate axes.
Find the distance between the foci of the ellipse.
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To find the distance between the foci of the ellipse, we can use the properties of ellipses and the given information. TargetPayandBenefits
First, let's denote the coordinates of the center of the ellipse as (h, k), where h represents the horizontal shift and k represents the vertical shift. Since the ellipse is tangent to the x-axis at (a, 0), we know that the distance between the center and the x-axis is a, which means k = a.
Since the ellipse is also tangent to the y-axis at (0, b), we know that the distance between the center and the y-axis is b, which means h = b.
Now, we have the center coordinates as (a, a). The distance between the foci of an ellipse can be calculated using the formula c = √(a^2 - b^2), where c represents the distance between the center and each focus.
Substituting h = b = a into the formula, we get c = √(a^2 - a^2) = √0 = 0.
Therefore, the distance between the foci of the given ellipse is 0.
